But the reasoning would be faulty. The line _ff_↓{1} only _touches_ the curve, it does not coincide with an element of the curve. Also at the point _b_↓{1} the pressure has a certain definite value, and there is no change. At the corresponding point _b_↓{11} the volume also has a certain definite value, and there is no change. There can therefore be no rate of variation. The value of the tangent does not give us a measure of the rate of variation: it gives us the _limit_ to the rate of variation, when the pressure is changing in the immediate vicinity of _b_↓{1}. We must stick to the notion of a pressure change in the _immediate vicinity_ of _b_↓{1}. What do we mean by “immediate vicinity”? We mean that we are thinking of a range of pressure-values in which the particular pressure-value _b_↓{1} is contained, but not as an end-point. We mean also that we choose a definite standard of approximation to the value _b_↓{1}, so that any pressure-value within our interval differs from _b_↓{1} by _less_ than this standard of approximation. It means further that, no matter how small is the number representing this standard of approximation, _any_ pressure-value within the interval will differ from _b_↓{1} by less than this number. This is what we really mean when we say that the interval we are thinking about is an “infinitely small one.” Now corresponding to this interval of pressure-values in the immediate vicinity of _b_↓{1}, there will be an interval of volume-values in the immediate vicinity of _b_↓{11}, and, as before, any one of these volume-values will differ from _b_↓{11} by less than any number representing a standard of approximation to _b_↓{11}. We then find the point on the curve corresponding to both _b_↓{1} and _b_↓{11}, that is b, and we draw the line _ff_↓{1}, and find the tangent of the angle which this line makes with _op_. The value of this tangent is the _limit_ of the rate of variation of the volume of the gas when the pressure undergoes a change in the immediate vicinity of _b_↓{1}. “Rate of variation” is a function of the argument “pressure.” This function has the limit _l_ for a value of its argument _b_↓{1}, when, as the argument varies in the immediate vicinity of _b_↓{1}, the value of the function approximates to _l_ within _any standard whatever_ of approximation.[36] [36] If the reader does not understand this, he should read Whitehead’s “Introduction to Mathematics.” He should read this book in any case.